Abstract
The presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system. We derive a bound on the speed with which observables of open quantum systems evolve. This speed limit divides into Mandalestam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, generalizing both to open quantum systems. By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds to the speed of evolution. We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits, and that a preferred basis of \emph{speed operators} serves to completely characterize the observables that saturate the speed limits. We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.
Highlights
How quickly can the expectation value hAi of an observable change as a quantum system evolves in the presence of an environment? What properties of the system allow for fast evolution of a physical quantity? How sensitive is the speed of an observable’s evolution to the effects of an environment? We probe these questions by deriving speed limits on observables, i.e., uncertainty relations that bound the rate of change of hAi
We derived speed limits on expectation values of observables for a quantum system evolving under arbitrary differentiable dynamics
This division of dynamics in terms of coherent and incoherent contributions was crucial to deriving upper bounds on speed that are tighter than those implied by the quantum Cramer-Rao bound, where we exploited the fact that the quantum Cramer-Rao bound is loose when applied to a particular estimator
Summary
We probe these questions by deriving speed limits on observables, i.e., uncertainty relations that bound the rate of change of hAi. Our main results rely on discriminating the “quantumlike” coherent contributions and the “classical-like” incoherent contributions to the evolution of an open quantum system. Mandelstam and Tamm first derived a bound on the speed for quantum systems evolving unitarily under a Hamiltonian H [1] They proved that dhAi dt ≤ 2ΔAΔH; ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ΔA ≔ hA2i − hAi2 and ΔH are the standard deviations of the observable and the Hamiltonian, respectively (units are such that ħ 1⁄4 1). While bounds on τ⊥ provide information about the fastest evolving Hermitian operators, they may not reflect the dynamics of experimentally relevant physical observables [49] This situation is exacerbated by the fact that the speed limits of different metrics can vary significantly [10,13]. In the Appendixes, we present derivations of these results
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